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Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Karpeyev, D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Mills, R.T., Munson, T., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H. and Zhang, H. (2020) PETSc Users Manual. Technical Report ANL-95/11 Revision 3.13, Argonne National Laboratory.

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• TITLE: Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

  AUTHORS: Andrea Brugnoli, Ghislain Haine, Anass Serhani, Xavier Vasseur

  KEYWORDS: Port-Hamiltonian Systems, Partial Differential Equations, Boundary Control, Structure-Preserving Discretization, Finite Element Method

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.9 No.6, June 18, 2021

  ABSTRACT: We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.